Theorem funopabeq 5924

Description: A class of ordered pairs of values is a function. (Contributed by NM, 14-Nov-1995.)

Assertion

Ref	Expression
funopabeq	|- Fun { <. x , y >. | y = A }

Distinct variable groups:   x, y   y, A

Allowed substitution hint:   A( x)

Proof of Theorem funopabeq

Step	Hyp	Ref	Expression
1		funopab 5923	. 2 |- ( Fun { <. x , y >. | y = A } <-> A. x E* y y = A )
2		moeq 3382	. 2 |- E* y y = A
3	1, 2	mpgbir 1726	1 |- Fun { <. x , y >. | y = A }

Syntax hints:   = wceq 1483   E*wmo 2471   {copab 4712   Fun wfun 5882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-fun 5890

This theorem is referenced by:  funopab4  5925